It's currently happening to "Western cultures", which are breeding below the replacement level and being supplanted by people in parts of Africa, Asia, and to a lesser extent South America.

However, many Western countries still have growing populations because of immigration, and while those people retain some of their culture, they also adopt some of ours. So I don't think "Western culture" is in danger of dying out. Plus we export it everywhere thanks to the Internet and TV, film, and books.

White people as a race seem to have their days numbered, but I think an inevitable part of globalization is people are mixing to such an extent now that we'll eventually all be some shade of tan. One less thing to divide us.